Study of Relative Risk Indexes

Decomposing Portfolio Value-at-Risk: A General Analysis

Winfried G. Hallerbach *)

Associate Professor, Department of Finance Erasmus University Rotterdam POB 1738, NL-3000 DR Rotterdam The Netherlands phone: [phone] facsimile: +31.10.408 9165 e-mail: [e-mail] http://www.few.eur.nl/few/people/hallerbach/ final version: October 15, 2002 forthcoming in The Journal of Risk 5/2, Febr. 2003

*) I’d like to thank Michiel de Pooter and Haikun Ning for excellent programming assistance. I appreciate the critical remarks and helpful comments of Jan Annaert, Phelim Boyle, Stephen Figlewski, Philippe Jorion (the editor), Ton Vorst, an anonymous referee and participants of the Northern Finance Association Meeting in Calgary. Of course, any remaining errors are mine.

Abstract

A variety of methods is available to estimate a portfolio’s Value-at-Risk. Aside from the overall VaR there is an apparent need for information about marginal VaR, component VaR and incremental VaR. Expressions for these VaR metrics have been derived under the restrictive normality assumption. In this paper we investigate these VaR concepts in an elliptical world and in a general distribution-free (simulation) setting, and show how they can be estimated.

Keywords: Value-at-Risk, marginal VaR, component VaR, incremental VaR, nonnormality, non-linearity, simulation

JEL classification: C13, C14, C15, G10, G11

1.

Introduction

Value-at-Risk (VaR) is defined as a one-sided confidence interval on potential portfolio losses over a specific horizon. Interest in such a diagnostic metric can be traced back to Edgeworth [1888] but the developments in this field were really spurred by the release of RiskMetrics™ by J.P.Morgan in October 1994. An intensive and still growing body of research focuses on estimating a portfolio’s VaR and various analytical or simulation-based methods have been developed (see for example Duffie & Pan [1997] and Jorion [2001] for an

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